Finite Group Databasebeta

← #27271598

Subgroups — #27271598

Order 13568.

334 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275785C211yesmaximal, characteristic
2#20272647C211yesmaximal, characteristic, complemented
3C53×#1393C211yesmaximal, characteristic, complemented
4C22×#11719922—21no—
5C2×#13133214C411yescharacteristic, complemented
6C2×#13133137—21no—
7#16810378C411yescharacteristic, complemented
8#16810349—21no—
9C2×#99×C53—21no—
10C53×#390C2211yescharacteristic
11C22×D212—21no—
12C22×D212—41no—
13C2×#11719922—42no—
14C2×#11719922—42no—
15C2×#11719922—41no—
16C2×#11719922—41no—
17C22×#88322—21no—
18#13133214—42no—
19#13133137—42no—
20C22⋊#88322—41no—
21C22×D4×C53D411yescharacteristic
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C22⋊C4×C53—41no—
24C2×C22⋊C4×C53—41no—
25C2×C42×C53C2×C411yescharacteristic
26#99×C53—22no—
27#99×C53—41no—
28C24×C53—21no—
29C23×D53—41no—
30C2×D4×C53—21no—
31C2×D4×C53—41no—
32C2×D4×C53—41no—
33C2×D4×C53D811yescharacteristic
34C2×D4×C53SD1611yescharacteristic
35C22×C4×C53—21no—
36C22×C4×C53—21no—
37C22×C4×C53—81no—
38C22×C4×C53C22⋊C411yescharacteristic
39C2×D212—22no—
40C2×D212—41no—
41C2×D212—41no—
42C2×D212—41no—
43C2×D212—41no—
44C2×D212—41no—
45C2×D212—42no—
46C2×D212—42no—
47C2×D212—82no—
48C2×D212—82no—
49C4⋊C4×C53—41no—
50C22⋊C4×C53—42no—
51C22⋊C4×C53—82no—
52C22⋊C4×C53—81no—
53C42×C53—41no—
54#11719922—44no—
55#11719922—44no—
56C2×#88322—22no—
57C2×#88322—41no—
58C2×#88322—41no—
59C2×#88322—81no—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53—81no—
65C23×C53—81no—
66C23×C53#9911yescharacteristic
67C22×D53—42no—
68C22×D53—82no—
69C22×D53—82no—
70C22×D53—81no—
71C22×D53—81no—
72D4×C53—41no—
73D4×C53—41no—
74D4×C53—81no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—81no—
82C2×C4×C53—82no—
83C2×C4×C53—161no—
84C2×C4×C53—161no—
85C2×C4×C53#10211yescharacteristic
86C2×C4×C53#10411yescharacteristic
87D212—42no—
88D212—42no—
89D212—81no—
90D212—81no—
91D212—84no—
92D212—84no—
93#88322—42no—
94#88322—161no—
95#7402—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—81no—
101C22×C53—81no—
102C22×C53—81no—
103C22×C53—81no—
104C22×C53—161no—
105C22×C53#32711yescharacteristic
106C22×C53AΣL(1,𝔽16)11yescharacteristic
107C22×C53#35311yescharacteristic
108C2×D53—84no—
109C2×D53—82no—
110C2×D53—162no—
111C2×D53—161no—
112C2×D53—161no—
113C4×C53—21no—
114C4×C53—21no—
115C4×C53—42no—
116C4×C53—81no—
117C4×C53—162no—
118#2011—531no—
119#1393D5311yescharacteristic, complemented
120#1318—531no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—81no—
124C2×C53—81no—
125C2×C53#134311yescharacteristic
126C2×C53#140611yescharacteristic
127C2×C53#140811yescharacteristic
128D53—162no—
129C22×D8—1061no—
130C2×#127—531no—
131C2×#102—1061no—
132C2×#99—21no—
133#390C2×D5311yescharacteristic
134#336—531no—
135#324—1061no—
136C53#740211yescharacteristic, complemented, Sylow
137C22×D4—1061no—
138C22×D4—2121no—
139C22×D4C22⋊D5311yescharacteristic
140C2×D8—2122no—
141C2×D8—2122no—
142C2×D8—2121no—
143C2×D8—2121no—
144C22×C8—1061no—
145#127—2122no—
146C2×C4⋊C4D21211yescharacteristic
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C42C4×D5311yescharacteristic
150#102—2122no—
151#99—22no—
152#99—41no—
153C22⋊C8—2121no—
154C24—21no—
155C24—2121no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—1062no—
160C2×D4—4242no—
161C2×D4—4242no—
162C2×D4—2121no—
163C2×D4—2121no—
164C2×D4—2121no—
165C2×D4—2122no—
166C2×D4—2122no—
167C2×D4—2121no—
168C2×D4—2121no—
169C2×D4#1171992211yescharacteristic
170C2×D4#1171992411yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—81no—
174C22×C4#1171992111yescharacteristic
175D8—2124no—
176D8—2124no—
177C2×C8—1062no—
178C2×C8—4241no—
179C2×C8—2121no—
180C2×C8—2121no—
181C4⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—82no—
184C22⋊C4—81no—
185C42—41no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—41no—
190C23—81no—
191C23—81no—
192C23—4242no—
193C23—4242no—
194C23—4241no—
195C23—4241no—
196C23—2122no—
197C23#1313313111yescharacteristic
198D4—41no—
199D4—41no—
200D4—81no—
201D4—4241no—
202D4—4241no—
203D4—4244no—
204D4—4244no—
205D4—2122no—
206D4—2122no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—161no—
216C2×C4—161no—
217C2×C4#1313313711yescharacteristic
218C2×C4#1313314511yescharacteristic
219C8—8481no—
220C8—2122no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—161no—
230C22—8482no—
231C22—8481no—
232C22—8481no—
233C22—4244no—
234C22—4242no—
235C22#1681035611yescharacteristic
236C22#1681040911yescharacteristic
237C22#1681041511yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—81no—
242C4—162no—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—81no—
247C2—8482no—
248C2#2027271111yescharacteristic
249C2#2027287111yescharacteristic
250C2#2027287711yescharacteristic
subgroupG/H
centerC22#16810415
commutatorC2×C42×C53C2×C4
Frattini#390C2×D53
FittingC53×#1393C2
socleC22×C53#353
prime radical#20275785C2
cosocleC53×#390C22
hypercenter#1393D53
hypocenterC53#7402
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1393

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20272647C2semidirect
3. C53×#1393C2semidirect
5. C2×#13133214C4semidirect
7. #16810378C4semidirect
119. #1393D53semidirect
136. C53#7402semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202757852
#202726472
C53×#13932
#740253